#134100

order	512
nilpotency class	4
isoclinism	isoclinic class of #133889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

874 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,21,25,17)(12,26)(13,18)(14,19,22,37)(15,20)(16,23,34,28)(24,38)(27,33)(29,36)(30,35,39,40)(31,32);(2,5)(3,7)(9,11,15,25)(10,16,20,34)(12,22,24,14)(13,28,31,23)(17,32,21,18)(19,36,37,33)(26,35,38,40)(27,30,29,39);(9,12,15,24)(10,17)(11,22,25,14)(13,29,31,27)(16,32)(18,34)(19,26)(20,21)(23,30,28,39)(33,40)(35,36)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41296);